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Quantum-field-theoretical approach to phase-space techniques:

Symmetric Wick theorem and multitime Wigner representation.

L.I.Plimak∗

Institut fur Quantenphysik, Universitat Ulm, 89069 Ulm, Germany.

M.K.OlsenARC Centre of Excellence for Quantum-Atom Optics, School of Physical Sciences,

University of Queensland, Brisbane, Qld 4072, Australia.

(Dated: May 11, 2018)

In this work we present the formal background used to develop the methods used in earlier worksto extend the truncated Wigner representation of quantum and atom optics in order to addressmulti-time problems. The truncated Wigner representation has proven to be of great practical use,especially in the numerical study of the quantum dynamics of Bose condensed gases. In these cases,it allows for the simulation of effects which are missed entirely by other approximations, such as theGross-Pitaevskii equation, but does not suffer from the severe instabilities of more exact methods.The numerical treatment of interacting many-body quantum systems is an extremely difficult task,and the ability to extend the truncated Wigner beyond single-time situations adds another powerfultechnique to the available toolbox. This article gives the formal mathematics behind the developmentof our “time-Wigner ordering” which allows for the calculation of the multi-time averages which arerequired for such quantities as the Glauber correlation functions which are applicable to bosonicfields.

PACS numbers: 02.50.Ey,03.65.Sq,05.10.Gg,05.40.-a

I. INTRODUCTION

This paper is a formal corollary to the recent Ref. [1],where the truncated Wigner representation of quantumoptics [2] was extended to multitime problems. (For adiscussion of operator orderings, such as normal, time-normal and symmetric, we refer the reader to the classictreatise of Mandel and Wolf [3].) The goal of Ref. [1] wasto develop a practical computational tool, while formalanalyses were reduced to the bare necessities. In thispaper we give proper justification to the formal tech-niques underlying Ref. [1]. The functional techniquesused here are to a large extent borrowed from Vasil’ev[4]; they were first outlined in preprint [5]. However, in[5] a number of important results (notably, continuityof time-symmetrically ordered operator products) wereoverlooked.Important for the putting the results of this paper in

perspective is the connection between phase-space tech-niques and the so-called real-time quantum field theory(for details and references see [6]). According to [6], theKeldysh rotation underlying the latter is a generalisa-tion of Weyl’s ordering to Heisenberg operators. Thisgeneralisation is nothing but the aforementioned time-symmetric ordering. This paper could thus equally betitled, “Phase space approach to real-time quantum fieldtheory”.For physical motivation and literature we refer the

∗Present address: Max Born Institute for Nonlinear Optics and

Short Pulse Spectroscopy, Division A1, 12489 Berlin, Germany. E-

mail : Lev.Plimak@mbi-berlin.de.

reader to the introduction to [1]. Here we only brieflytouch upon the more recent developments. The trun-cated Wigner representation has found its main util-ity for the investigation and numerical modelling ofBose-Einstein condensates (BEC), where the presenceof a significant third-order nonlinearity makes the exactpositive-P method unstable except for very short times[7], although it is still also used in quantum optics. Overthe last few years it has been used for an increasing num-ber of investigations, with the most relevant mentionedin what follows.

The ability to include the effects of initial quantumstates other than coherent was first shown numerically fortrapped BEC molecular photoassociation in three papersby Olsen and Plimak [8], Olsen [9], and Olsen, Bradley,and Cavalcanti [10], using methods later published byOlsen and Bradley [11] to sample the required quantumstates. Johnsson and Hope calculated the multimodequantum limits to the linewidth of an atom laser [12].The quantum dynamics of superflows in a toroidal trap-ping geometry were treated by Jain et al. [13]. Ferris etal. [14] investigated dynamical instabilities of BEC at theband edge in one-dimensional optical lattices, while Hoff-mann, Corney, and Drummond [15] made an attempt tocombine the truncatedWigner and positive-P representa-tions into a hybrid method for Bose fields. Corney et al.[16] used the truncated Wigner to simulate polarisationsqueezing in optical fibres, a system which is mathemat-ically analogous to BEC. The mode entanglement andEinstein-Podolsky paradox were numerically investigatedin the process of degenerate four-waving mixing of BECin an optical lattice by Olsen and Davis [17] and Ferris,Olsen, and Davis [18]. Also analysing BEC in an opti-

2

cal lattice, Shrestha, Javanainen, and Rusteokoski [19]looked at the quantum dynamics of instability-inducedpulsations. Midgley et al. performed a comparison ofthe truncated Wigner with the exact positive-P repre-sentation and a Hartree-Fock-Bogoliubov approximatemethod for the simulation of molecular BEC dissocia-tion [20], concluding that the truncated Wigner repre-sentation was the most useful in practical terms. Thispractical usefulness has been demonstrated in studies ofBEC interferometry [21], domain formation in inhomo-geneous ferromagnetic dipolar condensates [22], vortexunbinding following a quantum quench [23], a reverseKibble-Zurek mechanism in two-dimensional superfluids[24], the quantum and thermal effects of dark solitons inone-dimensional Bose gases [25], the quantum dynamicsof multi-well Bose-Hubbard models [26, 27], and analysisof a method to produce Einstein-Podolsky-Rosen statesin two-well BEC [28]. Along with its continuing use inquantum optics, the above examples demonstrate thatthe truncated Wigner representation is an extremely use-ful approximation method, allowing for the numericalsimulation of a number of processes for which nothingelse is known to be as effective.

The task in [1] was split naturally in two. Firstly,we constructed a path-integral approach in phasespace, which we called multitime Wigner representa-tion. Within this approach, truncated Wigner equationsemerged as an approximation to exact quasi-stochasticequations for paths. Secondly, we developed a way ofbringing time-symmetric products of Heisenberg oper-ators [1, 5], expressed by the path integral, to time-normal order [3]. The respective relations, called gener-alised phase-space correspondences , originate in Kubo’sformula for the linear response function [29, 30]. How-ever, the properties of time-symmetric operator products[1, 5] were formulated without proof. Details of the limit-ing procedure defining the path integral were ignored asirrelevant within the truncated Wigner approximation.

The logic of the present paper is best understood bydrawing an analogy with our Ref. [31]. In [31], our goalwas to extend normal-ordering-based approaches of quan-tum stochastics beyond quartic Hamiltonians. This pa-per is an attempt to extend the techniques of Ref. [31] fur-ther, this time to the Weyl-ordering-based method of Ref.[1]. Particulars aside, in [31] we applied causal [31], or re-sponse [32–34], transformation to Perel-Keldysh diargamseries [35, 36] for the system in question. The result wasa Wyld-type [37] diagram series, which we called causalseries . These kind of diagram techniques are well knownin the theory of classical stochastic processes [4, 38, 39].It is therefore not surprising that we could reverse engi-neer the causal series resulting in a stochastic differentialequation (SDE) for which this series was a formal solu-tion. For quartic (collisional) interactions the result isthe well-known positive-P representation of quantum op-tics. For other interactions, finding an SDE in the truemeaning of the term is not always possible because of thePawula theorem [40]. In such cases the causal series may

be only approximated by a stochastic difference equation(S∆E) in discretised time. Attempts to simulate emerg-ing S∆Es numerically have not been encouraging [31, 41].This is the primary reason why in Ref. [1] we resorted toapproximate methods.

The key point of [31] is a formal link existing be-tween conventional quantum field theory (Schwinger-Perel-Keldysh’s closed-time-loop formalism) and conven-tional quantum optics (time-normal operator orderingand positive-P representation). The existence of sucha link was first pointed out by one of the present au-thors in [42]. For the purposes of this paper, both back-grounds are inadequate or missing and have to be builtfrom scratch. On the quantum-field-theoretical side, wedevelop a formal framework of “symmetric Wick the-orems” expressing time-ordered operator products bysymmetrically-ordered ones. Similar to Wick’s theoremproper, this allows us to construct a “symmetric” vari-ety of Perel-Keldysh diagram series with corresponding“symmetric” propagators. On the quantum-optical side,we introduce what we called a multitime Wigner repre-sentation generalising Weyl’s ordering to multitime av-erages of Heisenberg operators. The necessary “flavour”of causal transformation can then be borrowed from [5].The link to time-normal ordering is then based on a fun-damental link between commutators of Heisenberg oper-ators for different times and the response properties ofquantum fields [31–34, 42]. All our results apply to non-linear quantum systems with, to a large extent, arbitraryinteraction Hamiltonians.

The existence of symmetric Wick theorems for thechange of time ordering to symmetric may be of inde-pendent interest for a wider audience than that for whichthis paper is primarily intended. It is for this wider audi-ence that in the appendix we present the generalisation ofsymmetric Wick theorems to arbitrary quantised fields.

The glue that holds the paper together is the concept oftime-symmetric ordering of Heisenberg operators. Con-tinuing the analogy with [31], this ordering replaces thetime-normal operator ordering on which [31] was built.On the quantum-field-theoretical side, we express “sym-metric” propagators by the retarded Greens function.The induced restructuring of “symmetric” Perel-Keldyshseries automatically turns them into perturbative se-ries for quantum averages of time-symmetrically orderedproducts of Heisenberg operators. On the quantum-optical side, time-symmetric ordering appears as a nat-ural generalisation of the conventional symmetric order-ing to Heisenberg operators. Quantum field theory andquantum optics are not equal here. Time-symmetric or-dering emerges in quantum field theory as a fully specifiedformal concept. At the same time, there does not seemto be any way of guessing, save deriving, this conceptfrom within conventional phase-space techniques with-out a reference to Schwinger’s closed-time-loop formalism[43].

The paper is organised as follows. In sections II andIII, two formal concepts are prepared for later use. In sec-

3

tion II we define time-symmetric ordering of Heisenbergoperators and prove its most important properties: real-ity, continuity, and the fact that for free fields it reducesto conventional symmetric ordering. In section III weprove “symmetric Wick theorems” generalising Wick’stheorem proper from normal to symmetric ordering. Insection IV, we introduce a functional framework and de-rive closed perturbative relations with symmetric order-ing of operators. The multitime Wigner representationof an arbitrary bosonic system is formulated in sectionV. In section VIA we construct a representation of time-symmetric operator averages by phase-space path inte-grals. Section VII presents a discussion of causal regu-larisation needed to make our analyses mathematicallydefined. The problem of reordering Heisenberg operatorsis discussed in section VIII. In the appendix, we extendsymmetric Wick theorems to arbitrary quantised fields.

II. THE TIME-SYMMETRIC ORDERING

A. The oscillator basics

We introduce the common pair of bosonic creation andannihilation operators,

[

a, a†]

= 1, (1)

and the usual free oscillator Hamiltonian,

H0 = ω0a†a. (2)

We use units where h = 1. The symmetric (Weyl) or-dering of the creation and annihilation operators is con-veniently defined in terms of the operator-valued charac-teristic function

χ(η, η∗) = eη∗a+ηa†

, (3)

so that, by definition

∂m+nχ(η, η∗)

∂ηn∂η∗m

∣

∣

∣

η=0= Wama†n. (4)

For simplicity, we consider a single-mode case; multi-mode cases are recovered formally by attaching mode in-dex to all quantities, cf. section IVA and the appendix.Operator orderings act mode-wise, so that extension tomany modes is straightforward.The time-dependent creation and annihilation opera-

tors are defined as,

a(t) = ae−iω0t, a†(t) = a†eiω0t. (5)

Formally, they are Heisenberg field operators with re-spect to the free Hamiltonian, e.g.,

i∂ta(t) =[

a(t), H0(t)]

, (6)

where H0(t) is H0 in the interaction picture,

H0(t) = ω0a†(t)a(t). (7)

We shall term a(t), a†(t) free-field, or interaction-picture,operators, because this is the role they play in real prob-lems with interactions. Symmetric ordering is extendedto free fields postulating that,

Wa(t1) · · · a(tm)a†(tm+1) · · · a†(tm+n)

= e−iω0(t1+···+tm−tm+1−···−tm+n) Wama†n. (8)

Furthermore, adding an arbitrary, and, in general, time-dependent, interaction term to H0,

H = H0 + HI, (9)

allows us to introduce the pair of the Heisenberg fieldsoperators proper,

A(t) = U†(t)aU(t), A†(t) = U†(t)a†U(t), (10)

where the evolution operator is defined through theSchrodinger equation,

idU(t)dt

= HU(t), limt→−∞

U(t) = 11. (11)

Interaction picture is introduced in the usual way bysplitting the evolution operator in two factors,

U(t) = U0(t)UI(t), (12)

where U0(t) is the evolution operator with respect to H0,

idU0(t)

dt= H0U0(t), lim

t→−∞U0(t) = 11, (13)

and UI(t) obeys the equation,

idUI(t)

dt= HI(t)UI(t), lim

t→−∞UI(t) = 11, (14)

with HI(t) being HI in the interaction picture,

HI(t) = U†0 (t)HIU0(t). (15)

The reader should have noticed that we adhere to cer-tain notational conventions. Calligraphic letters are re-served for evolution and Heisenberg operators. Plainletters denote Schrodinger and interaction-picture oper-ators, which are in turn distinguished by the absenceor presence of the time argument, respectively. TheSchrodinger operator a becomes a(t) in the interaction

picture and A(t) in the Heisenberg picture,

a(t) = U†0(t)aU0(t),

A(t) = U†(t)aU(t) = U†I (t)a(t)UI(t),

(16)

and similarly for other operators.We note that, with unspecified interaction, Heisenberg

operators are in essence placeholders. By specifying in-teraction one ipso facto specifies all Heisenberg opera-tors.

4

B. The time-symmetric operator products

Throughout the paper we make extensive use of theconcept of time-symmetric product of the Heisenberg op-erators [1]. We define it by the following recursive pro-cedure. For a single operator the ordering is irrelevant,

T WA(t) = A(t), T WA†(t) = A†(t). (17)

Furthermore, if P[>t] is a time-symmetric product whereall time arguments are larger than t, then

T WP[>t] A(t) =1

2

[

P[>t], A(t)]

+,

T WP[>t] A†(t) =1

2

[

P[>t], A†(t)]

+,

(18)

where [· · · ]+ stands for the anticommutator,

[

X , Y]

+= X Y + YX . (19)

This allows one to built a time-symmetric product ofany number of factors, by applying (18) in the or-der of decreasing time arguments (increasing in [5] isa typo). The result are nested anticommutators, (with

Xk = A, A†, k = 1, · · · , N)

T WX1(t1)X2(t2) · · · Xn(tn) =1

2N−1

×[

· · ·[[

X1(t1), X2(t2)]

+, X3(t3)

]

+, · · · , XN (tN )

]

+,

t1 > t2 > · · · > tN . (20)

Equations (18) and (20) imply that all time argumentsin a time-symmetric product are different. Extension tocoinciding time arguments may be given by continuity,see below.For two factors we find plain symmetrised combina-

tions,

T WA(t1)A(t2) =1

2

[

A(t1)A(t2) + A(t2)A(t1)]

,

T WA(t1)A†(t2) =1

2

[

A(t1)A†(t2) + A†(t2)A(t1)]

,

T WA†(t1)A†(t2) =1

2

[

A†(t1)A†(t2) + A†(t2)A†(t1)]

,

(21)

where the order of times does not matter. For three andmore factors it already does, e.g.,

T WA(t1)A(t2)A†(t3) =1

4

[[

A(t1), A(t2)]

+, A†(t3)

]

+

=1

4

[

A(t1)A(t2)A†(t3) + A(t1)A†(t3)A(t2) + A†(t3)A(t1)A(t2) + A(t2)A(t1)A†(t3)

+ A(t2)A†(t3)A(t1) + A†(t3)A(t2)A(t1)]

, t1 > t2 > t3, (22)

as opposed to

T WA(t1)A(t2)A†(t3) =1

4

[[

A(t1), A†(t3)]

+, A(t2)

]

+

=1

4

[

A(t1)A(t2)A†(t3) + A(t1)A†(t3)A(t2) + A†(t3)A(t1)A(t2) + A(t2)A(t1)A†(t3)

+ A(t2)A†(t3)A(t1) + A†(t3)A(t2)A(t1)]

, t1 > t3 > t2. (23)

In these formulae, the crossed-out terms are those ab-sent in time-symmetric operator products but occuringin symmetrised ones.

C. General properties of time-symmetric products

Equation (20) expresses a time-symmetrically orderedproduct of N factors as a sum of 2N−1 products of fieldoperators which differ in the order of factors. In all theseproducts the time arguments first increase then decrease(whereas in crossed-out terms in (22) and (23) the earli-est operator is in the middle). This kind of time-ordered

structure is characteristic of Schwinger’s closed-time-loopformalism (see Refs. [36, 43] and sections III C and IV be-low), so we shall talk of Schwinger (time) sequences andof the Schwinger order of factors in Schwinger products .It is easy to see that all possible Schwinger products ap-pear in the sum. Indeed, the earliest time in a Schwingersequence can only occur either on the left or on the right.That recursions (18) generate all Schwinger products canthen be shown by induction, starting from pair productsfor which this is obvious. We may thus give an alterna-tive definition of the time-symmetric product:

5

The time-symmetric product of N factors is 1/2N−1

times the sum of all possible Schwinger products ofthese factors.

This definition of the time-symmetric product is illus-trated in Fig. 1, where the 2N−1 Schwinger products com-prising it are visualised as 2N−1 distinct ways of placingthe operators on the so-called C-contour [35, 36, 43, 44].The latter travels from t = −∞ to t = +∞ (the forwardbranch) and then back to t = −∞ (reverse branch). Theoperators are imagined as positioned on the C-contour;each operator may be either on the forward or reversebranch, cf. Fig. 1. The order of operators on the C-contour determines the order of factors in a particularSchwinger product (from right to left, to match Eq. (58)below).Definition of a time-symmetric product as a sum of all

Schwinger products makes obvious the reality property,

[

T W(· · · )]†

= T W(· · · )†. (24)

This follows from the trivial fact that if a particular se-quence of times is in a Schwinger order, the reverse se-quence is also in a Schwinger order. Furthermore, visu-alisation in Fig. 1 helps us to prove the most importantproperty of the time-symmetric products: their continu-ity at coinciding time arguments. Assume that a pair oftimes t, t′ can change their mutual order but both stayeither earlier or later in respect of all other times, cf. Fig.1. Assume also that placing of all times on the C-contour

t1 t2 t3 t4 t5 t8 t9 t11

t10t7

t6^Pr

^lP P>

^

−tt

FIG. 1: Graphical representation of Schwinger products pro-duced by the recursion (18). This particular example im-

plies N = 11 operators Xk(tk), k = 1, · · · , N , with t1 <

t2 < · · · < tN . The operators are imagined as positionedon the C-contour travelling from t = −∞ to t = +∞ (theforward branch) and then back to t = −∞ (reverse branch).Each operator may be either on the forward or on the re-verse branch. The order of operators on the C-contour deter-mines the order of factors in a particular Schwinger product(from right to left, to match Eq. (58)). So, the way of plac-ing operators shown by dark circles corresponds to the prod-uct X1X3X6X10X11X9X8X7X5X4X2, where time argumentsare omitted for brevity. Placing of the latest time (t11) doesnot affect the order of operators, so we put it arbitrarily onthe forward branch. With all alternative placings of operators(light circles) we recover 2N−1 distinct Schwinger products.For the operator placing shown by dark circles, the quanti-ties occuring in equation (25) are Pl = X1(t1)X3(t3), Pr =

X5(t5)X4(t4)X2(t2), P> = X10(t10)X11(t11)X9(t9)X8(t8), t6 =min(t, t′), and t7 = max(t, t′).

except t, t′ is fixed. If t, t′ are not the latest times, weare left with four ways of placing them on the C-contourresulting, up to the overall coefficient, in four terms (with

X , Y = A, A†)

Q(t, t′) = PlX (t)P>Y(t′)Pr + PlY(t)P>X (t′)Pr

+

{

PlX (t)Y(t′)P>Pr + PlP>Y(t′)X (t)Pr, t < t′,

PlY(t′)X (t)P>Pr + PlP>X (t)Y(t′)Pr, t > t′.

(25)

Here, the operator product P> comprises all operator fac-

tors with time arguments larger than t, t′, Pr comprisesoperators with time arguments less than t, t′ placed onthe forward branch of the C-contour, and Pl comprisesoperators with time arguments less than t, t′ placed onthe reverse branch of the C-contour (see Fig. 1). Then,

limtրt′

Q(t, t′)− limtցt′

Q(t, t′)

= Pl

[

X (t), Y(t)]

P>Pr − PlP>

[

X (t), Y(t)]

Pr = 0,

(26)

because the commutator here may only be a c-number(or zero); this holds also in a multimode case. If t, t′ are

the latest times, then P> = 11 and the freedom of placingt, t′ reduces to two terms,

Q(t, t′) = Pl

[

X (t)Y(t′) + Y(t)X (t′)]

Pr, (27)

which is independent of the order of the times t, t′. Con-tinuity of the time-symmetric products has thus beenproven. Note that this spares us the necessity to specifytheir values at coinciding time arguments.

D. Equivalence of the time-symmetric and

symmetric ordering of the free-field operators

We now prove that, when applied to free fields, thetime-symmetric ordering is just a fancy way of redefiningconventional symmetric ordering defined by Eqs. (4) and(8),

T Wa(t1) · · · a(tm)a†(tm+1) · · · a†(tm+n)

= Wa(t1) · · · a(tm)a†(tm+1) · · · a†(tm+n)

= e−iω0(t1+···+tm−tm+1−···−tm+n) Wama†n, (28)

so that the order of times here in fact does not matter.At first glance, equations (22) and (23) serve as counter-examples. From Eq. (22) we get,

T Wa(t1)a(t2)a†(t3) =

e−iω0(t1+t2−t3)

2

×[

a2a† + a†a2]

, t1 > t2 > t3, (29)

6

as contrasted by the formula obtained from Eq. (23),

T Wa(t1)a(t2)a†(t3) =

e−iω0(t1+t2−t3)

4

×[

a2a† + a†a2 + 2aa†a]

, t1 > t3 > t2. (30)

Neither result coincides with the symmetrically-orderedproduct following from (4),

Wa(t1)a(t2)a†(t3) =

e−iω0(t1+t2−t3)

3

×[

a2a† + a†a2 + aa†a]

. (31)

However by using the commutational relation (1) it iseasy to verify that

1

2

[

a2a† + a†a2]

=1

4

[

a2a† + a†a2 + 2aa†a]

=1

3

[

a2a† + a†a2 + aa†a]

= Wa2a†. (32)

To prove (28) in general we note that the recursionprocedure defining the time-symmetric product may bestarted from the unity operator,

T W11 = 11. (33)

Then,

T WA(t) =1

2

[

11, A(t)]

+= A(t),

T WA†(t) =1

2

[

11, A†(t)]

+= A†(t),

(34)

in obvious agreement with (17). Furthermore, we can

replace 11 in (33) by χ(η, η∗) given by Eq. (3) and thenapply the recursion (18) starting from the latest time.With the end limit η = 0 this replacement does not af-fect the resulting time-symmetric products. So, assumingthat t′ < t,

T Wa(t) =1

2

[

χ(η, η∗), a(t)]

+|η=0,

T Wa(t)a†(t′) =1

4

[[

χ(η, η∗), a(t)]

+, a†(t′)

]

+|η=0.

(35)

Similar relations hold for larger number of factors. Wenow transform them using the standard phase-space cor-respondences. Recalling that

eη∗a+ηa†

= eηa†

eη∗ae|η|

2/2 = eη∗aeηa

†

e−|η|2/2 (36)

we find,

∂χ(η, η∗)

∂η=

(

a† + η∗/2)

χ(η, η∗)

= χ(η, η∗)(

a† − η∗/2)

,

∂χ(η, η∗)

∂η∗= (a− η/2)χ(η, η∗)

= χ(η, η∗) (a+ η/2) .

(37)

Pairwise combining these relations and multiplying themby suitable time exponents we obtain

1

2[χ(η, η∗), a(t)]+ = e−iω0t

∂χ(η, η∗)

∂η∗,

1

2

[

χ(η, η∗), a†(t)]

+= eiω0t

∂χ(η, η∗)

∂η.

(38)

This allows us to rewrite Eqs. (35) as

T Wa(t) = e−iω0t ∂χ(η, η∗)

∂η∗

∣

∣

∣

η=0,

T Wa(t)a†(t′) = e−iω0(t−t′) ∂2χ(η, η∗)

∂η∂η∗

∣

∣

∣

η=0.

(39)

For an arbitrary number of factors we have,

T Wa(t1) · · · a(tm)a†(tm+1) · · · a†(tm+n)

= e−iω0(t1+···+tm−tm+1−···−tm+n) ∂m+nχ(η, η∗)

∂ηn∂η∗m

∣

∣

∣

η=0.

(40)

By virtue of (4) this is another form of (28).To conclude this paragraph we note that Eq. (28) is

not as straightforward as the corresponding relation forthe normal ordering:

:a†(tm+1) · · · a†(tm+n)a(t1) · · · a(tm):

= e−iω0(t1+···+tm−tm+1−···−tm+n):a†nam:. (41)

Here setting the free-field operators in the normal orderipso facto sets the creation and annihilation operators inthe normal order, while in Eq. (28) the symmetric orderis only recoved on rearranging the operators using (1).Since the commutator contains dynamical informationone may say that relation (41) is purely kinematical whileequation (28) is dynamical.

E. The time-symmetric averages

The time-symmetric ordering of free-field operators de-livers us “the best of both worlds.” On the one hand,it is just symmetric ordering in disguise. On the otherhand, the Schwinger products of which it is build areconsistent with such “big guns” as Schwinger’s closed-time-loop formalism. The time-symmetric ordering ofthe free-field operators correctly “guesses” certain fun-damental structures underlying the interacting quantumfield theory, making it an irreplaceable bridging conceptwhen deriving perturbative relations with the symmetricordering.In practice, it is convenient to eliminate symmetric or-

dering altogether by reexpressing all information aboutthe initial state of the system directly in terms of time-symmetric averages. We introduce the Wigner distribu-tion in the standard way by

〈χ (η, η∗)〉 =∫

d2α

πW (α, α∗) eη

∗α+ηα∗

, (42)

7

where the quantum averaging is over the initial state ofthe system. The symmetric averages characterising theinital state may then be written as

⟨

Wama†n⟩

=

∫

d2α

πW (α, α∗)αnα∗n. (43)

By making use of this relation, the result of averagingEq. (28) takes the form

⟨

T Wa(t1) · · · a(tm)a†(tm+1) · · · a†(tm+n)⟩

=

∫

d2α

πW (α, α∗)α(t1) · · ·α(tm)

× α∗(tm+1) · · ·α∗(tm+n), (44)

where

α(t) = αe−iω0t, α∗(t) = α∗eiω0t. (45)

If W (α, α∗) > 0, this is nothing but stochastic momentsof the random c-number field α(t), which is specified byits value at t = 0 (initial condition) being distributed inaccordance with the probability distribution W (α, α∗).For nonpositive W (α, α∗) this interpretation holds by re-placing probability by quasi-probability.

III. SYMMETRIC WICK THEOREMS

A. Time ordering of operators

The time ordering places operators from right to leftin the order of increasing time arguments, e.g.,

TW+ A(t)A†(t′) = θ(t− t′)A(t)A†(t′)

+ θ(t′ − t)A†(t′)A(t), (46)

and similarly for a larger number of factors. By defi-nition, bosonic operators under the time ordering com-mute. The notation TW

+ as distinct from T+ used in

[31] emphasises that the TW+ -ordering implies a different

specification for coinciding time arguments. In [31] thisspecification was by normal ordering, while, not quite un-expectedly, in this paper we imply symmetric ordering.In [31], such specification was enforced by the causal reg-ularisation which was part of the dynamical approach.A similar (but different) regularisation scheme is part ofthe dynamical approach in this paper, cf. section VII.Till that section we suppress all specifications related tooperator orderings for coinciding time arguments. Thenotation TW

+ should thus be regarded a placeholder forthe concept full meaning of which will be made clear insection VII.

B. Symmetric Wick theorem for the time ordering

We now prove a generalisation ofWick’s theorem to thesymmetric ordering. As a major simplification, results

of the previous section allow us to formulate and provethe symmetric Wick theorem as a relation between thetime-ordered and time-symmetrically ordered operatorsproducts:

A time-ordered product of interaction-picture oper-ators equals a sum of time-symmetrically orderedproducts with all possible symmetric contractions,including the term without contractions.

The symmetric contraction is defined, in obvious analogyto Wick’s theorem proper, as

−iGW++(t− t′) = TW

+ a(t)a†(t′)− T Wa(t)a†(t′). (47)

One can use here the symmetric ordering in place of thetime-symmetric one, but we follow our general principleof eliminating the W -ordering from considerations. Thecumbersome notation we use for the symmetric contrac-tion will be explained in section III C below. For theoscillator,

−iGW++(t) = ε(t)

[

a(t), a†(0)]

= ε(t)e−iω0t, (48)

where ε(t) is the odd stepfunction,

ε(t) =1

2[θ(t)− θ(−t)] . (49)

The symmetric Wick theorem obviously holds for oneand two operators; in the latter case it coincides with Eq.(47). A general proof follows by induction. Assume thesymmetric Wick theorem has been proven up to a certainnumber of factors and consider a time-ordered productwith one “spare” factor. We choose this spare factor asthe earliest one, which is thus on the right of the timeordered product. Let it be a(t). The whole time-orderedproduct may then be written as

TW+ a†(t1) · · · a†(tm)a(tm+1) · · · a(tm+n)a(t) = P a(t),

(50)

where we have introduced a notation for the productwithout the “spare” factor

P = TW+ a†(t1) · · · a†(tm)a(tm+1) · · · a(tm+n). (51)

We wish to have the spare factor symmetrically on eitherside of P so we write

P a(t) =1

2

[

P , a(t)]

++

1

2

[

P , a(t)]

. (52)

The commutator is easily calculated

1

2

[

P , a(t)]

=1

2

m∑

k=1

P ′k

[

a†(tk), a(t)]

, (53)

where P ′k is P without the factor a†(tk); note that P ′

kremain time-ordered. For t < tk,

1

2

[

a†(tk), a(t)]

= −iGW++(t− tk), (54)

8

GW GW

GW

GW

t

t

FIG. 2: The C-contour and the contractions. The contrac-tions are directed from creation to annihilation, cf. Eq. (59).

and we find

TW+ a†(t1) · · · a†(tm)a(tm+1) · · · a(tm+n)a(t)

=1

2

[

P, a(t)]

+− i

m∑

k=1

P ′kG

W++(t− tk). (55)

We now use the induction assumption and expand P andall P ′

k according to the symmetric Wick theorem. Byvirtue of (18) the anticommutator in (55) is then a sumof time-symmetric products. It gives us the sum of termswhere contractions exclude a(t). The sum in (55) de-liveres the terms where a(t) is involved in a contraction.It is straightforward to verify that all terms reguired bythe symmetric Wick theorem are recovered this way, eachoccuring only once.The case when the earliest term is a†(t) is treated by

simply swapping in the above a ↔ a†. Equation (54) isreplaced by

1

2

[

a(tk), a†(t)

]

= −iGW++(tk − t), t < tk (56)

which indeed holds, cf. Eq. (48). The symmetric Wicktheorem has thus been proven. We note that the reg-ularisation we mentioned after Eq. (46) results, in par-ticular, in GW

++(0) = 0 without mathematical ambiguity.Hence the caveat of Wick’s theorem, that “no contrac-tions should occur between operators with equal time ar-guments,” also applies to the symmetric Wick theorem.

C. Symmetric Wick theorem for the

closed-time-loop ordering

If the initial state of the system is not vacuum, closedequations of motion cannot be written in terms of av-erages of the time-ordered operator products only, cf.,e.g., Ref. [44]. The necessary type of ordering was in-troduced, rather indirectly, in Schwinger’s seminal paper[43], and later applied to developing diagram techniquesfor nonequilibrium nonrelativistic quantum problems byKonstantinov and Perel [35]. Extension to relativisticproblems was given by Keldysh [36] under whose namethe approach became known. We note that the concept ofthe closed time loop and the related operator ordering ismuch more general than the Keldysh diagram techniquescommonly associated with it. As was shown in paper

[31], it underlies such a phase-space concept as the time-normal ordering of Glauber and Kelly and Kleiner. Inthis paper we investigate the relation between the closed-time-loop and the symmetric orderings.

The closed-time-loop operator ordering is commonlydefined as an ordering on the so-called C-contour whichcan be seen in Figs. 1 and 2. The C-contour travels fromt = −∞ to t = +∞ (forward branch) and then back tot = −∞ (reverse branch). The operators are formally as-signed an additional binary argument which distinguishesoperators on the forward branch from those on the re-verse branch. “Earlier” and “later” are then generalisedto match the travelling rule along the C-contour. All op-erators on the forward branch are by definition “earlier”than those on the reverse branch and go to the right.Among themselves, the operators on the forward branchare set from right to left in the order of increasing time ar-guments (TW

+ -ordered), while those on the reverse branchare set from right to left in the order of decreasing timearguments (TW

− -ordered). Using that Hermitian conju-

gation inverts the time order of factors, the TW− -ordering

may be defined as

TW− (· · · ) =

[

TW+ (· · · )†

]†. (57)

The closed-time-loop-ordered operator product may thusbe alternatively defined as a double-time-ordered prod-uct,

TW− P− TW

+ P+, (58)

where P± are two independent operator products. In

terms of the C-contour, the operators in P+ are visualisedas positioned on the forward branch of the C-contour,and those in P−—on the reverse branch. Factors in adouble-time-ordered product are always arranged in aSchwinger sequence, cf. the opening paragraph of sectionII C; for specifications pertaining to coinciding time ar-guments we refer the reader to section VII. In this paperwe employ Eq. (58) as a primary definition and use theconcept of closed time loop only for illustration purposes.

Extension of the symmetric Wick theorem to the dou-ble time ordering employs the linear order of the C-contour. A formal generalisation of the symmetric Wicktheorem to an arbitrary linearly ordered set may be foundin the appendix. Hori’s form of the symmetric Wick the-orem given by Eq. (67) below, of which the derivation isour goal, is in fact a particular case of the general struc-tural relation (A.14). Nonetheless we believe that thedirect proof we present here will benefit the reader.

The symmetric Wick theorem is generalised to thedouble-time ordering by making the symmetric contrac-tion dependent on the positioning of the contracted pairin the double-time-ordered product (or, which is thesame, on the C-contour, cf. Fig. 2). As a result we recover

9

four contractions:

−iGW++(t− t′) = TW

+ a(t) a†(t′)− T Wa(t)a†(t′),

−iGW−−(t− t′) = TW

− a(t) a†(t′)− T Wa(t)a†(t′),

−iGW−+(t− t′) = TW

− a(t)TW+ a†(t′)− T Wa(t)a†(t′),

−iGW+−(t− t′) = TW

− a†(t′)TW+ a(t)− T Wa(t)a†(t′).

(59)

Unlike in Ref. [31], there are four nonzero symmetric con-tractions, not three. To make the genesis of the contrac-tions clearer we retained the orderings also where theyare redundant, e.g., TW

− a(t)TW+ a†(t′) = a(t)a†(t′). The

contraction −iGW++(t) is exactly that defined by (47) ex-

plaining the notation. As c-number kernels, −iGW++(t)

and −iGW−−(t) are coupled by Hermitian conjugation,

while −iGW+−(t) and −iGW

−+(t) are Hermitian:

−iGW−−(t− t′) =

[

− iGW++(t

′ − t)]∗,

−iGW+−(t− t′) =

[

− iGW+−(t

′ − t)]∗,

−iGW−+(t− t′) =

[

− iGW−+(t

′ − t)]∗.

(60)

In obtaining this we used (24) as well as the similar re-lation for the double-time ordering,

[

TW− P− TW

+ P+

]†= TW

− P†+ TW

+ P†−, (61)

cf. Eq. (57). For the oscillator,

−iGW++(t) = iGW

−−(t) = ε(t)e−iω0t,

−iGW−+(t) = iGW

+−(t) =1

2e−iω0t,

(62)

cf. Eq. (48).With the contractions defined by (59) the symmetric

Wick theorem reads:

A double-time-ordered product of interaction-picture operators equals a sum of time-symmetrically ordered products with all possiblesymmetric contractions, including the term withoutcontractions.

This includes the time-ordered products as a specialcase. Importantly, the generalisation to the double-time-ordering makes the symmetric Wick theorem invariantunder Hermitian conjugation. This follows from Eqs.(24) and (61), supplemented by the observation that Her-mitian conjugation turns every Schwinger time sequenceinto a reversed sequence, so that arguments of all contrac-tions must change sign. Equations (60) show that this isexactly the effect the compex conjugation has on the con-tractions, including the replacement GW

++ ↔ GW−−. It is

therefore only necessary to generalise the inductive stepin the above proof to the case when the “spare” operatoris the earliest one under the TW

+ ordering. Again, let usfirstly assume that this operator is a(t). Equation (52)

holds with P now being a double-time-ordered product,

P = TW− P− TW

+ P+. (63)

Equation (53) applies as well, by assuming that P ′k

“knows” whether a†(tk) originates from P− or from P+.The critical observation is that the contractions dependin fact only on the visual order of times, so that, remem-bering that t < tk,

1

2

[

a†(tk), a(t)]

= −iGW++(t− tk) = −iGW

+−(t− tk).

(64)

With this observation equation (55) is replaced by

TW− P− TW

+ P+a(t) =1

2

[

P, a(t)]

+

− i∑

tk∈P−

P ′kG

W+−(t− tk)− i

∑

tk∈P+

P ′kG

W++(t− tk),

(65)

where∑

tk∈P+means summation over all k such that

a†(tk) originates from P+, and similarlly for P−. Thecase when the earliest operator under the TW

+ -ordering

is a†(t) is treated similarly, the “critical observation” inthis case being that, for tk > t,

1

2

[

a(tk), a†(t)

]

= −iGW++(tk − t) = −iGW

−+(tk − t).

(66)

This completes the inductive step of the proof. The sym-metric Wick theorem has thus also been proven for thedouble time ordering.

D. Functional form of the symmetric Wick theorem

One technical tool in Ref. [31] was Hori’s form ofWick’s theorem expressing it as an application of a func-tional differential operator, cf. Eq. (18) in [31]. Except forthe redefinition of contractions, the symmetric Wick the-orem coincides with Wick’s theorem proper. This makesthe functional form of Wick’s theorem equally applicableto the symmetric Wick theorem. All we need is to rede-fine the differential operator ∆C given by Eq. (19) in [31]as

∆WC

[

δ

δa+,

δ

δa+,

δ

δa−,

δ

δa−

]

= −i∑

c,c′=±

∫

dtdt′GWcc′(t− t′)

δ2

δac(t)δac′(t′), (67)

where a±(t), a±(t) are four independent c-number fields.Square brackets signify functionals. The symmetric con-tractions were defined by Eqs. (59). With these replace-ments we find Hori’s form of the symmetric Wick theo-

10

rem,

T−P−

[

a, a†]

T+P+

[

a, a†]

= T W

{

exp∆WC

[

δ

δa+,

δ

δa+,

δ

δa−,

δ

δa−

]

× P−

[

a−, a−]

P+

[

a+, a+]

∣

∣

∣

a→a

}

. (68)

Here, P±[·, ·] are two independent functionals of time-dependent fields, and a → a stands for the replacementof the c-numbers by operators,

a±(t) → a(t), a±(t) → a†(t). (69)

Once again, exact meaning to Eq. (68) will be assignedby regularisations in section VII.

IV. THE CLOSED-TIME-LOOP FORMALISM

A. The model

All definitions and majority of formulae in this and thefollowing sections do not depend on the system Hamil-tonian and can be applied to any bosonic system. Tobe specific, and to maintain connections with our pre-vious paper [1] we demostrate our techniques for a one-dimensional Bose-Hubbard (1D BH) chain. This systemcombines simplicity of the formal model with practicalrelevance: starting from the seminal paper by Jaksch etal. [45] the BH model evolved into a standard approachto cold atoms trapped in optical lattices. The fact thata particular relation below is specific to this model willalways be stated explicitly. The simplest version of the1D BH model describes a chain of N anharmonic oscilla-tors with Josephson’s tunneling (“hopping”) between thenearest neighbours,

H =

N∑

k=1

[

ω0a†kak +

κ

2a†2k a2k

− J(a†kak+1 + a†k+1ak)]

, (70)

where nk = a†kak, and a†k, ak is the standard creation-

annihilation pair for the k-th site,[

ak, a†k′

]

= δkk′ . Theindices are understood modulo N , aN+1 = a1; in otherwords, we consider a ring rather than a chain. As usualwe break Hamiltonian (70) into the “free” and “interac-tion” Hamiltonians,

H = H0 + HI, H0 =

N∑

k=1

H0k =

N∑

k=1

ω0nk. (71)

The free-field and the Heisenberg operators, ak(t), a†k(t)

and Ak(t), A†k(t) are defined in detailed analogy to sec-

tion IIA. We exclude hopping from the free Hamiltonian

making our reasoning most easlily adaptable to arbitrarymulti-mode bosonic systems. In particular this allows theanalyses in the previous two sections to be generalisedsimply by applying them mode-wise.

B. Perel-Keldysh Green functions

We now introduce the formal framework that will serveus throughout the rest of the paper. A brief histori-cal introduction into the Schwinger-Perel-Keldysh closed-time-loop formalism was given at the beginning of sectionIII C. The basic quantity in this formalism is a double-time-ordered Green function⟨

TW− Ak1

(t1) · · · Akm(tm)A†

km+1(tm+1) · · · A†

km+m(tm+m)

× TW+ Ak′

1(t′1) · · · Ak′

n(t′n)A†

k′n+1

(t′n+1) · · · A†k′n+n

(t′n+n)⟩

,

(72)

where m, m, n, n ≥ 0 are arbitrary integers. Specifica-tions of Eq. (72) for equal time arguments are postponedtill section VII, cf. the remarks at the end of section III B.A convenient interface to the whole assemblage of func-tions (72) is their generating, or characteristic, functional

ΞW[

ζ−, ζ−, ζ+, ζ+

]

=⟨

TW− exp

(

− iζ−A− iζ−A†)

× TW+ exp

(

iζ+A+ iζ+A†)

⟩

, (73)

where ζk±(t), ζ†k±(t) are four arbitrary c-number func-

tions per mode. Notationally we treat them, as well asthe mode operators, as vectors. We follow the sign con-ventions of Ref. [34]. To emphasise the structural side ofour analyses we employ a condensed notation,

xy =

N∑

k=1

∫

dt xk(t)yk(t),

xXy =

N∑

k,k′=1

∫

dtdt′ xk(t)Xkk′ (t− t′)yk′(t′).

(74)

Here, xk(t) and yk(t) are arbitrary functions andXkk′ (t− t′) is an arbitrary kernel; for q-numbers the or-der of factors matters.

C. Closed perturbartive relations

The main advantage of the functional framework isthat the universal structural part of the perturbative cal-culations can be expressed as a small set of closed pertur-bative relations . This was demonstrated in [31] for thenormal ordering, and will be demonstrated in this paperfor the symmetric ordering. We wish to construct a per-turbative formula the generating functional (73). By the

11

same means as equation (24) was found in Ref. [31] werewrite Eq. (73) in the interaction picture as

ΞW[

ζ−, ζ−, ζ+, ζ†+

]

=⟨

TW− exp

(

−iζ−a− iζ−a† + iLW

I

[

a, a†])

× TW+ exp

(

iζ+a+ iζ+a† − iLW

I

[

a, a†])⟩

. (75)

Here, LWI is a functional of two arguments,

LWI

[

a, a†] =

∫

dtHWI

(

a(t), a†(t))

, (76)

where the c-number function HWI (·, ·) is the symmetric

form of the interaction Hamiltonian,

HI = WHWI

(

a, a†). (77)

The last equation is written in the Schrodinger picture.Note that

UI = TW+ exp

(

− iLWI

[

a, a†]) (78)

is the interaction-picture S-matrix. Equation (77) is gen-eral and holds for any interaction. For the Bose-Hubbardchain,

HI =

N∑

k=1

[

κ

2a†2k a2k − J

(

a†kak+1 + a†k+1ak

)

]

= WN∑

k=1

[

κ

2a†2k a2k − κa†kak +

κ

4

− J(

a†kak+1 + a†k+1ak

)

]

≡ HWI , (79)

cf. Eq. (70). In deriving this we have used that, for theharmonic oscillator,

W{

a†a}

=1

2

(

a†a+ aa†)

,

W{

a†2a2}

=1

6

(

a†2a2 + a†aa†a+ a†a2a†

+ aa†2a+ aa†aa† + a2a†2)

.

(80)

The reason why the symmetric form of the interactionmust be used is exactly that why in [31] we had to usethe normal form. The key property of the T+-ordering inRef. [31] is that it does not affect the single-time normallyordered operator products, so that, in particular,

T+

{

:HI(t):}

= :HI(t):. (81)

We have the same consistency between TW+ and HW

I (t):

TW+

{

HWI (t)

}

= HWI (t). (82)

Since the order of operators under TW+ is fully decided by

this ordering, the only way to prevent it from redefiningthe interaction is to put the latter into symmetric form.Operators may be eliminated from Eq. (75) altogether

by, firstly, applying the symmetric Wick theorem (68) soas to bring the operator construct under the quantumaveraging to a (time-)symmetrically ordered form, and,secondly, using a multimode generalisation of Eq. (44) toexpress the average. The result of this transformationreads

ΞW[

ζ−, ζ−, ζ+, ζ+

]

=

∫

d2α1

π· · · d

2αN

πW

(

α,α∗)

{

exp∆WC

[

δ

δa+,

δ

δa+,

δ

δa−,

δ

δa−

]

× exp(

− iζ−a− − iζ−a− + iLWI

[

a−, a−

]

+ iζ+a+ + iζ+a+ − iLWI

[

a+, a+

]

)

}

∣

∣

∣

a→α. (83)

In the above,

α(t) = αe−iω0t, α∗(t) = α∗eiω0t, (84)

W (α,α∗) is the multimode Wigner function (cf. Eq.(44)), and a → α stands for the replacement,

a±(t) → α(t), a±(t) → α∗(t). (85)

Not to be confused by Eq. (83), note that the func-tional differential operation within the curly brackets is

applied under the condition that a±(t), a±(t) are fourarbitrary c-number vector functions. These functionsare then replaced pairwise by α(t),α∗(t), which dependonly on the initial condition α. This replacement turnsthe functional expression in curly brackets into a func-tion of the initial condition α, making averaging over theWigner function W (α,α∗) meaningful.

12

D. Formal solution to the quantum-statistical

response problem

Following Kubo, we add a source term to the Hamil-tonian (70),

H ′ = H −N∑

k=1

[

a†ksk(t) + aks∗k(t)

]

. (86)

The Heisenberg operators corresponding to H ′ will bedenoted as A′

k(t). Similar to (73), we introduce a char-acteristic functional for the double-time-ordered averagesof A′

k(t), A′†(t):

ΞW[

ζ−, ζ−, ζ+, ζ+

∣

∣s, s∗]

=⟨

TW− exp

(

− iζ−A′ − iζ−A

′†)

× TW+ exp

(

iζ+A′ + iζ+A

′†)

⟩

. (87)

The condensed notation we use here was defined by (74).With a taste for the paradoxical, the message of this

section may be formulated as, the quantum response prob-lem does not exist, because the information on the re-sponse properties of the system is already present in thecommutators of the field operator [32–34]. Formally, thisis expressed by the following relation between the char-acteristic functionals,

ΞW[

ζ−, ζ−, ζ+, ζ†+

∣

∣s, s∗]

= ΞW[

ζ− + s, ζ− + s∗, ζ+ + s, ζ†+ + s∗

]

. (88)

This formula is a trivial consequence of the closed per-turbative relation (75); it suffices to note that adding thesource term to the Hamiltonian results in the followingreplacement in Eq. (83),

LWI

[

a, a]

→ LWI

[

a, a]

− as− as∗. (89)

Equation (88) shows that all information one needs inorder to predict response of a quantum system to an ex-ternal source is already present in the Heisenberg fieldoperators defined without the source. A closer inspec-tion of Eq. (88) reveals that this information is “stored”in the commutators of the Heisenberg operators. For de-tails see Refs. [32–34].

V. THE WIGNER REPRESENTATION

A. Symmetric Wick theorem and causality in

phase space

Similar to [31], the starting point of the ensuing deriva-tion is understanding the causal structure of the symmet-ric contractions (59). We begin with the observation that

(

ω0 − id

dt

)

GW++(t) = δ(t). (90)

This characterises the symmetric contraction GW++(t) as

a Green function of the c-number equation

(

ω0 − id

dt

)

ak(t) = sk(t) + sk(t). (91)

To spare us future redefinitions we wrote (91) as anequation for the phase-space trajectories ak(t). We alsobroke the total source into the given external sourcesk(t) and the self-action source sk(t) accounting for theinteractions. In general, sk(t) is quasi-stochastic andfield-dependent. Note that here we work with a quasi-stochastic equation in the classical phase space, unlike inRef. [31] where stochastic equations in the phase space ofdoubled dimension were considered. As a result, here thereader encounters complex-conjugate field pairs insteadof independent field pairs as in ref. [31].

While GW++(t) is obviously one of Green functions of

Eq. (91), it is wrong as far as natural causality is con-cerned. In phase space as well as in classical mechan-ics physics are associated with causal response definedthrough the retarded Green function,

GR(t) = iθ(t)e−iω0t. (92)

This quantity also obeys equation (90) but, unlikeGW

++(t), is retarded. Causal solutions of Eq. (91) arespecified by replacing it by the integral equation

ak(t) = αk(t) +

∫

dt′GR(t− t′)[

sk(t′) + sk(t

′)]

, (93)

with the condition ak(t) → αk(t) as t → −∞. By analogywith Ref. [31] we expect the in-field α(t) to be definedby the multimode generalisation of Eq. (44). That is, theinitial condition for Eqs. (91) is defined by the initial stateof the system. Consistency of these assumptions andthe way Eq. (91) is linked to quantum averages remainsubject to verification.

B. The causal transformation

As in Ref. [31] we proceed by noticing that all contrac-tions may be expressed by GR(t). Simply by trial anderror it is easy to get

GW++(t) = −GW

−−(t) =1

2

[

GR(t) +G∗R(−t)

]

,

GW−+(t) = −GW

+−(t) =1

2

[

GR(t)−G∗R(−t)

]

.

(94)

Similar to [31], we are looking for variables which wouldbring ∆W

C given by (67) to the form

∆WC =

δ

δaGR

δ

δξ∗+

δ

δa∗G∗

R

δ

δξ. (95)

13

We use here condensed notation defined by Eq. (74).Equation (95) takes us to the substitution

a+(t) = a(t)− iξ(t)

2, a+(t) = a∗(t)− iξ∗(t)

2,

a−(t) = a(t) +iξ(t)

2, a−(t) = a∗(t) +

iξ∗(t)

2.

(96)

These relations imply that

a∗+(t) = a−(t), a∗

−(t) = a+(t). (97)

Under this condition the symmetric Wick theorem (68)remains valid. It is also consistent with the substitution(85), which in variables a(t), ξ(t) becomes,

a(t) → α(t), a∗(t) → α∗(t), ξ(t) → 0. (98)

In the below we assume that Eq. (97) holds.

C. Response in the Wigner representation

Continuing the analogy with Ref. [31], we impose thecondition,

− iζ−(t)a−(t)− iζ−(t)a−(t) + iζ+(t)a+(t)

+ iζ+(t)a+(t) = ζ(t)a∗(t) + ζ∗(t)a(t)

+ ξ(t)σ∗(t) + ξ∗(t)σ(t), (99)

on the linear form in Eq. (83). This results in anothersubstitution, this time in functionals (73) and (87):

ζ+(t) = σ(t)− iζ(t)

2, ζ+(t) = σ∗(t)− iζ∗(t)

2,

ζ−(t) = σ(t) +iζ(t)

2, ζ−(t) = σ∗(t) +

iζ∗(t)

2.

(100)

The inverse substitution reads,

ζ(t) = i[

ζ+(t)− ζ−(t)]

, ζ∗(t) = i[

ζ+(t)− ζ−(t)]

,

s(t) =1

2

[

ζ+(t) + ζ−(t)]

, s∗(t) =1

2

[

ζ∗+(t) + ζ

∗−(t)

]

.

(101)

showing that (100) is a genuine change of functional vari-ables. Similar to Eqs. (96) and (97), these relations im-pose conditions on the functional arguments,

ζ∗+(t) = ζ−(t), ζ∗

−(t) = ζ+(t). (102)

These conditions do not interfere with (96) and (97) serv-ing as characteristic functionals for the correspondingGreen function, nor with (100) being a one-to-one sub-stitution.

By definition, the generalised multitime Wigner rep-resentation emerges by applying substitution (100) tofunctionals (96) and (97). To start with, we note thatthe replacement,

ζ±(t) → ζ±(t) + s(t), ζ±(t) → ζ±(t) + s(t), (103)

cf. Eq. (88), in variables ζ(t),σ(t) becomes simply,

σ(t) → σ(t) + s(t), (104)

with variable ζ(t) unaffected. In variables ζ(t),σ(t) func-tionals “with and without the sources” are naturally ex-pressed by a single functional ΦW ,

ΦW[

ζ, ζ∗∣

∣σ,σ∗]

= ΞW

[

σ +iζ

2,σ∗ +

iζ∗

2,σ − iζ

2,σ∗ − iζ∗

2

]

,

ΦW[

ζ, ζ∗∣

∣σ + s,σ∗ + s∗]

= ΞW

[

σ +iζ

2,σ∗ +

iζ∗

2,σ − iζ

2,σ∗ − iζ∗

2

∣

∣

∣

∣

s, s∗]

.

(105)

We see that the functional variable σ(t) corresponds to the formal input of the system, which in turn is specified bythe formal c-number source added to the Hamiltonian. We emphasise formality of both concepts. Under macroscopicconditions, an external source may become a good approximation for a laser (say). Importantly, even in this case, itremains a phenomenological model for a complex quantum device.

D. Time-symmetric operator ordering

If σ(t) defines an input of the system, what does the output defined by variable ζ(t) stand for? Following [5], wepostulate that, in the Wigner representation, the formal output of a system is expressed by time-symmetric averages

14

of the Heisenberg field operators,

⟨

T W exp(

ζ∗A

′ + ζA′†)⟩

≡ ΦW[

ζ, ζ∗∣

∣s, s∗]

= ΞW

[

iζ

2,iζ∗

2,−iζ

2,−iζ∗

2

∣

∣

∣

∣

s, s∗]

. (106)

For the operator without the source,

⟨

T W exp(

ζ∗A+ ζA†

)⟩

≡ ΦW[

ζ, ζ∗∣

∣0, 0]

= ΞW

[

iζ

2,iζ∗

2,−iζ

2,−iζ∗

2

]

. (107)

Then,

⟨

T WAk1(t1) · · · Akm

(tm) A†km+1

(tm+1) · · · A†km+m

(tm+m)⟩

=

δm+mΞW

[

iζ

2,iζ∗

2,−iζ

2,−iζ∗

2

]

δζ∗k1(t1) · · · δζ∗km

(tm)δζkm+1(tm+1) · · · δζkm+m

(tm+m)

∣

∣

∣

∣

ζ=0

, (108)

and similarly for the primed operator.It is easy to prove that Eq. (108) agrees with the recursive definition of section II B. Explicitly,

ΞW

[

iζ

2,iζ∗

2,−iζ

2,−iζ∗

2

]

=

⟨

TW− exp

(

1

2ζA† +

1

2ζ∗

A

)

TW+ exp

(

1

2ζA† +

1

2ζ∗

A

)⟩

, (109)

For simplicity we assume that all times on the LHS of (108) are different. We can then also assume that

ζk(t) =m+m∑

l=1

ζl,k(t) = ζ1,k(t) + ζ′k(t), (110)

where ζl,k(t) is nonzero only in close vicinity of tl, and that different ζl,k(t) do not overlap. Furthermore, if t1 is thesmallest time in (108), isolating in (109) the contribution linear in ζ

∗1 we can write

ΞW

[

iζ

2,iζ∗

2,−iζ

2,−iζ∗

2

]

∣

∣

∣

linear in ζ∗1

=

⟨(

1

2ζ∗1A

)[

TW− exp

(

1

2ζ ′A

† +1

2ζ′∗

A

)

TW+ exp

(

1

2ζ′A

† +1

2ζ ′∗

A

)]⟩

+

⟨[

TW− exp

(

1

2ζ′A

† +1

2ζ′∗

A

)

TW+ exp

(

1

2ζ ′A

† +1

2ζ′∗

A

)](

1

2ζ∗1A

)⟩

. (111)

The square brackets of the RHS of this formula delineatethe range to which the remaining time orderings are ap-plied. Clearly equation (111) is nothing but an elaborateform of the recursive definition (18). The case when the

“earliest” operator is one of the A†s follows by complexconjugation of (111).

VI. DYNAMICS IN PHASE-SPACE

A. Phase-space path integral

Physically, the most natural way of looking at the sys-tem is through time-symmetric averages of the field op-erator in the presence of the source. Their characteristicfunctional ΦW

[

ζ, ζ∗∣

∣s, s∗]

is given by Eq. (106). On the

one hand, it allows one full access to quantum propertiesof the system expressed by Perel-Keldysh Green func-tions (72). On the other hand, it has an obvious phase-space interpretation found by generalising Eq. (44) toHeisenberg operators. This equation may be written as,

⟨

T W exp(

ζ∗a+ ζa†)⟩ = exp

(

ζ∗α+ ζα∗)

, (112)

where, we remind,

ζ∗a =

N∑

k=1

∫

dtζ∗k (t)ak(t), ζ∗α =

N∑

k=1

∫

dtζ∗k (t)αk(t),

(113)

and similarly for other “products.” Functions αk(t) =αke

−iω0t depend on the initial condition α. The bar in

15

(112) symbolises quasi-averaging over the Wigner func-tion, cf. Eq. (44). Following the pattern of Eq. (112)we postulate the path-integral representation of the func-tional (106),

⟨

T W exp(

ζ∗A

′ + ζA′†)⟩

= exp(

ζa∗ + ζ∗a)

∣

∣

∣

s,s∗.

(114)

In this relation, a(t) is the c-number phase-space trajec-tory, and the bar denotes a path integral over these tra-jectories. We assume that trajectories obey the genericequation (93), making them dependent (conditional) onthe external source s(t) and on the initial condition α.The bar includes quasi-averaging over the initial condi-

tion and the own quasi-stochasticity of trajectories com-ing from the self-action source sk(t) in (93). For com-parison, in (112) the quasi-averaging is over the initialcondition while the way trajectories evolve in time is non-stochastic.

B. Closed perturbative relations in the Wigner

representation

Rewriting Eq. (83) in variables a(t), σ(t), ζ(t), andξ(t) defined by Eqs. (101) and (96), and replacing σ(t)by s(t), we obtain

⟨

T W exp(

ζ∗A

′ + ζA′†)⟩

=

∫

d2α1

π· · · d

2αN

πW

(

α,α∗)

{

exp

(

δ

δaGR

δ

δξ∗+

δ

δa∗G∗

R

δ

δξ

)

× exp(

ζa∗ + ζ∗a+ ξs∗ + ξ∗s+ SW[

ξ, ξ∗∣

∣a,a∗]

)

}∣

∣

∣

∣

a(t)→α(t),ξ(t)→0

, (115)

where

SW[

ξ, ξ∗∣

∣a,a∗]

= i

∫

dt

[

HWI

(

a(t) +iξ(t)

2,a∗(t) +

iξ∗(t)

2

)

−HWI

(

a(t)− iξ(t)

2,a∗(t)− iξ∗(t)

2

)]

. (116)

For better understanding of Eq. (115) the reader should reread the comments following Eq. (83).

C. Multitime-Wigner equations for the Bose-Hubbard model

Equations (115), (116) are general and hold for any interaction specified by its symmetrically ordered form HWI .

For the Bose-Hubbard chain we find (omitting time arguments in the integrand)

SW[

ξ, ξ∗∣

∣a,a∗]

= −N∑

k=1

∫

dt

{

κ[(

ξka∗k + ξ∗kak

)(

a∗kak − 1)]

− κ

4

(

ξ2kξ∗ka

∗k + ξ∗2k ξkak

)

− J(

ξ∗kak+1 + ξ∗k+1ak + ξka∗k+1 + ξk+1a

∗k

)

}

. (117)

As was shown in [31], Eq. (115) may be interpreted as asolution to the generic stochastic equation in phase spaceformulated in section VA. The random source in Eq. (91)is characterised by its averages conditioned on the “localfield” ak(t),

exp

N∑

k=1

∫

dt [ξ∗k(t)sk(t) + ξk(t)s∗k(t)]

∣

∣

∣

∣

a

= expSW[

ξ, ξ∗∣

∣a,a∗]

, (118)

so that SW[

ξ, ξ∗∣

∣a,a∗]

is a characteristic functional forthe cumulants of the random source. For the Bose-Hubbard model we have,

sk(t) = −κak(t)[

|ak(t)|2 − 1]

+ J[

ak+1(t) + ak−1(t)]

+ s(3)k (t), (119)

where the actual random contribution comes from thethird-order noise s

(3)k (t). It is specified by the conjugate

16

pair of nonzero cumulants,

s(3)k (t)s

(3)k′ (t′)s

(3)∗k′′ (t′′)

∣

∣

a

= δkk′δkk′′δ(t− t′)δ(t− t′′)κak(t)

2,

s(3)k (t)s

(3)∗k′ (t′)s

(3)∗k′′ (t′′)

∣

∣

a

= δkk′δkk′′δ(t− t′)δ(t− t′′)κa∗k(t)

2,

(120)

while all other cumulants of s(3)k (t) are zero (this allowed

us to specify the average in place of the cumulant). Whilethe Wigner function responsible for the initial conditionin (91) may be positive thus affording a statistical inter-pretation, the cubic noise is a purely pseudo-stochasticobject.

D. The truncated Wigner representation

Dropping the third-order noise turns the full Wignerrepresentation into the so-called truncated Wigner rep-resentation. In more traditional phase-space techniques,the truncated Wigner is found by dropping the third-order derivatives in the generalised Fokker-Planck equa-tion for the single-time Wigner distribution. In the ab-sence of losses, the corresponding Langevin equation isnon-stochastic. For the Hamiltonian (70), it coincideswith Eq. (91), where the source is given by (119) with-

out s(3)k . However simple and straightforward, the con-

ventional way of deriving the truncated Wigner leavesit unclear if it can be applied to any multitime quan-tum averages. In our approach here as well as in Ref. [1]the truncated-Wigner equations emerge as an approxima-tion within rigorous techniques intended for calculationof the time-symmetric averages of the Heisenberg oper-ators. The generalisation associated with extending thetruncated-Wigner equations to multitime averages is thushighly nontrivial and requires a new concept: the time-symmetric ordering of the Heisenberg operators. Theredoes not seem to be a way of guessing this concept fromwithin the conventional phase-space techniques.

VII. THE CAUSAL REGULARISATION

Strictly speaking, all relations derived so far are lead-ing considerations that require specifications. Two thingsneed to be warranted: that the functional form of thesymmetric Wick theorem conforms with the specificationof TW

+ as symmetric ordering for coinciding time argu-ments, and that the stochastic integral equation (93) isdefined mathematically. In Ref. [31], both problems weretaken care of “in a single blow” by the causal regularisa-tion [31, 41, 46, 47] of the retarded Green function GR(t),cf. Eq. (92). Namely, GR(t) should be replaced by a suf-ficiently smooth function while preserving its causal na-

ture, GR(t) ∼ θ(t). One may assume, for instance, that

GR(t) = iθ(t)(

1− e−Γt)m

e−iω0t, (121)

where the limit Γ → ∞ is implied. The regularised Greenfunction is zero at t = 0 and has m− 1 zero derivatives.Equation (121) is a toy version of the Pauli-Villars reg-ularisation used in the quantum field theory [48, 49] aspart of the common renormalisation procedure (cf. also[6]).

In this paper as well as in [31] the causal regularisationof GR(t) has a two-fold effect. Firstly, it assigns math-ematical sense to equations (91), (93), defining them asIto equations. In [31], this was in agreement with themore traditional approach based on pseudo-distributionsand generalised Fokker-Planck equations [50]. Further-more, with GR(t) regularised, equation (94) assures thatthe kernels GW

cc′(t) are smooth functions and

GWcc′(0) = 0 (122)

without any mathematical ambiguity. As a result, thesymmetric Wick theorem (68) leaves alone any productof operators with equal time arguments. Since the finalexpression in (68) is ordered symmetrically, the symmet-ric ordering is also enforced for any same-time producton the LHS of (68). Regularisation (121) applied to thesymmetric Wick theorem (68) thus indeed specifies thedouble-time-ordered product (58) so that operators withequal time arguments are ordered symmetrically.

An unwanted effect of regularisation is that symmet-ric ordering is also enforced for same-time groups of op-erators split between the TW

+ and TW− orderings. For

example, with regularisation,

TW− a†k′ (t

′)TW+ ak(t) = Wak(t)a

†k′ (t

′)

− δkk′

2

(

1− e−Γ|t−t′|)m

e−iω0|t−t′|. (123)

This problem was also encountered in [31], and the so-lution to it remains the same: the limit Γ → ∞ shouldalways preceed t′ → t. Then,

TW− a†k′ (t)T

W+ ak(t)

= limt′→t

limΓ→∞

TW− a†k′(t

′)TW− ak(t)

= Wak(t)a†k′(t)−

δkk′

2= a†k′(t)ak(t) (124)

as expected. The general recipe is to keep time argumentsof operators under the TW

+ and TW− orderings slightly

different, which is equivalent to ignoring regularisationof GW

+− and GW−+. This recipe implies that the double-

time-ordered averages in the limit Γ → ∞ are continuousfunctions of all time differences t+− t−, where t+ and t−are time arguments of an operator pair split between theTW+ and TW

− orderings. This is obviously consistent with

17

continuity properties of the GW+− and GW

−+ kernels,

limtց0

limΓ→∞

GW−+(t) = lim

tր0limΓ→∞

GW−+(t) = GW

−+(0),

limtց0

limΓ→∞

GW+−(t) = lim

tր0limΓ→∞

GW+−(t) = GW

+−(0),(125)

where GW−+ and GW

+− are unregularised kernels. Withthis amendment all specifications enforced by regularisa-tion only apply to operators under the time orderings. Infact, we only need to specify the TW

+ -ordering, defining itas time ordering for different times and symmetric order-ing for equal times. The TW

− -ordering remains definedby (57). We return to the problem of “unwanted effects”of regularisation in the next section.

VIII. THE REORDERING PROBLEM

Assume we know a way of calculating time-symmetricaverages of Heisenberg operators while the physics de-mand time-normal ones, or vice versa. For two-time av-erages this problem is solved by Kubo’s formula for thelinear response function [29, 30], cf. Refs. [1, 51]. Herewe consider a general approach to reordering Heisenbergoperators. Among other results we show that this alwaysreduces to considering a response problem of sorts.

It is instructive to compare the formulae relating thetime-symmetric and time-normal averages to the Perel-Keldysh Green functions (72):

ΦW[

ζ, ζ∗∣

∣s, s∗]

= ΞW

[

iζ

2+ s,

iζ∗

2+ s∗,

−iζ

2+ s,

−iζ∗

2+ s∗

]

, (126)

ΦN[

ζ, ζ∗∣

∣s, s∗]

= ΞN[

iζ + s, s∗, s,−iζ∗ + s∗]

. (127)

cf. section V. The functionals ΞW and ΞN are both char-acteristic ones for Green functions (72) but imply differ-ent specifications for coinciding time arguments: (126)implies symmetric ordering while (127) — normal order-ing. Such specifications only become of importance ifGreen functions (or their linear combinations) are con-sidered for coinciding time arguments, and otherwise canbe disregarded.

Mathematically, ΞW and ΞN coincide up to a singularpart . Hence, up to singular parts, the functionals ΦW

and ΦN differ only in a functional substitution appliedto Eq. (73). Time-symmetric averages appear with

η− =iζ

2+ s, η†− =

iζ∗

2+ s∗,

η+ =−iζ

2+ s, η†+ =

−iζ∗

2+ s∗,

(128)

while the time-normal ones require

η− = iζ + s, η†− = s∗,

η+ = s, η†+ = −iζ∗ + s∗.(129)

Both Eqs. (128) and (129) may be inverted, and it isstraightforward to show that

ΦW[

ζ, ζ∗∣

∣s, s∗]

= ΦN

[

ζ, ζ∗

∣

∣

∣

∣

s− iζ

2, s∗ +

iζ∗

2

]

,

ΦN[

ζ, ζ∗∣

∣s, s∗]

= ΦW

[

ζ, ζ∗

∣

∣

∣

∣

s+iζ

2, s∗ − iζ∗

2

]

.

(130)

In particular, for the time-normal and time-symmetric

averages defined without the source we have

ΦW[

ζ, ζ∗∣

∣0, 0]

= ΦN

[

ζ, ζ∗

∣

∣

∣

∣

− iζ

2,iζ∗

2

]

,

ΦN[

ζ, ζ∗∣

∣0, 0]

= ΦW

[

ζ, ζ∗

∣

∣

∣

∣

iζ

2,− iζ∗

2

]

.

(131)

These relations make it evident that the reordering prob-lem for the Heisenberg operators is indeed equivalent tothe response problem.We remind the reader that Eqs. (130) and (131) hold

only up to a singular part. All formulae for quantumaverages following from them are only valid for differenttime arguments and otherwise require specifications.We now consider an example which also illustrates how

to approach coinciding time arguments. We wish to ex-press the time-normal average

⟨

A†k(t)Ak′ (t′)

⟩

=δ2ΦN

[

ζ, ζ∗∣

∣0, 0]

δζk(t)δζ∗k′ (t′)

∣

∣

∣

ζ=0(132)

by the time-symmetric ones. Combining (132) with thesecond of Eqs. (131) we have

⟨

A†k(t)Ak′ (t′)

⟩

=⟨

T WA†k(t)Ak′ (t′)

⟩

+i

2

[

δ⟨

A′k′(t′)

⟩

δsk(t)− δ

⟨

A′†k (t)

⟩

δs∗k′ (t′)

]∣

∣

∣

∣

s=0

. (133)

In deriving this we used that

ΦW,N[

0, 0∣

∣s, s∗]

= 1, (134)

so that the derivatives purely by input arguments alwaysdisappear. In [1], Eq. (133) was derived by an ad hoc

18

method; here we show that the same relation follows fromthe general equations (131). For details on numericalimplementation of Eq. (133) see [1].Equation (133) is well behaved in the limit t′ → t. For

instance, let t′ > t. Then, by causality [33],

δ⟨

A†k(t)

⟩

δs∗k′ (t′)= 0. (135)

Using (21) we rewrite (133) as

δ⟨

A′k′(t′)

⟩

δs(t)

∣

∣

∣

s=0= i

⟨

[

Ak′ (t′), A†k(t)

]

⟩

, t′ > t. (136)

We have thus rederived Kubo’s famous relation for thelinear response function. Both sides here have a well-defined limit iδkk′ for t ր t′. Similarly,

δ⟨

A′†k (t)

⟩

δs∗k′(t′)

∣

∣

∣

s=0= −i

⟨

[

Ak′ (t′), A†k(t)

]

⟩

, t > t′, (137)

which is nothing but complex conjugation of (136) withthe replacement t ↔ t′.At the same time, Eq. (133) resists any attempt to

extend it to equal time arguments. Within the causalreqularisation,

δ⟨

A′k(t)

⟩

δsk(t)=

δ⟨

A′†k (t)

⟩

δs∗k(t)= 0, (138)

giving rise to a patently wrong result,

⟨

A†k(t)Ak(t)

⟩

=⟨

T WA†k(t)Ak(t)

⟩

. (139)

Conversely, making the response correction in (133)nonzero for t = t′ inevitably causes troubles with causal-ity under regularisation. This shows, in particular, thatthe “unwanted effects” of regularisation (see section VII)are unavoidable and cannot be eliminated by a betterregularisation scheme. For this reason we regard it as“good conduct” to treat all quantum averages as gener-alised functions, making the question of their values atcoinciding time arguments meaningless.

IX. CONCLUSIONS

We have developed a generalisation of the symmet-ric ordering to multitime problems with nonlinear inter-actions. This includes generalisation of the symmetric(Weyl) ordering to time-symmetric ordering of Heisen-berg operators, and of the renowned Wigner function toa path integral in phase-space. Among other results, con-tinuity of time-symmetric operator products has beenproven. A way of calculating time-normally-orderiedoperator products within the time-symmetric-ordering-based techniques has also been developed.

Acknowledgments

The authors are grateful to S. Stenholm, W. Schle-ich, M. Fleischhauer, and A. Polkovnikov for their com-ments on the manuscript. M.O. thanks the Institut furQuantenphysik at the Universitat Ulm for generous hos-pitality. L.P. is grateful to ARC Centre of Excellencefor Quantum-Atom Optics at the University of Queens-land for hospitality and for meeting the cost of his visitto Brisbane. This work was supported by the ProgramAtomoptik of the Landesstiftung Baden-Wurttembergand SFB/TR 21 “Control of Quantum Correlations inTailored Matter” funded by the Deutsche Forschungsge-meinschaft (DFG), Australian Research Council (GrantID: FT100100515) and a University of Queensland NewStaff Grant.

Appendix: Structural symmetric Wick theorem for a

pair of fields

In this appendix, we formulate an ultimate generali-sation of the symmetric Wick theorem. It appears tocover all imaginable cases of interest and, if necessary,can be extended to fermionic fields. Firstly, let τ be ageneralised time variable which belongs to some linearlyordered set with the succession symbol ≻. Of practicalrelevance are the time axis and the C-contour. Secondly,

let X(ξ, τ), ˆX(ξ, τ) be a pair of free-field operators de-fined in some Fock space, (in this appendix h 6= 1)

X(ξ, τ) =√h∑

κ

[

uκ(ξ)aκ(τ) + vκ(ξ)b†κ(τ)

]

,

ˆX(ξ, τ) =√h∑

κ

[

uκ(ξ)a†κ(τ) + vκ(ξ)bκ(τ)

]

.(A.1)

Here, ξ symbolises arguments of field operators excepttime; ξ can contain coordinate, momentum, spin indices,and so on. The index κ enumerates the modes of whichthe Fock space is built. With extentions to relativity orsolid state in mind, for each mode we define two pairs ofcreation and annihilation operators,

aκ(τ) = e−iϕκ(τ)aκ, a†κ(τ) = eiϕκ(τ)a†κ,

bκ(τ) = e−iϕκ(τ)bκ, b†κ(τ) = eiϕκ(τ)b†κ.(A.2)

The phases ϕκ(τ) are c-number functions of ‘time;” notethat Eq. (A.2) employs one phase for “particles and an-tiparticles” in each mode. The stationary operators obeystandard commutational relations,

[

aκ, a†κ′

]

=[

bκ, b†κ′

]

= δκκ′ , (A.3)

otherwise mode operators pairwise commute. The c-number functions uκ(ξ), vκ(ξ), uκ(ξ), and vκ(ξ) are re-placed in particular problems by suitable solutions ofa single-body equation. In nonrelativistic problems,vκ(ξ) = vκ(ξ) = 0. For our purposes here all c-number

19

functions in Eqs. (A.1) and (A.2) are regarded as arbi-trary.Specification of the quantities occuring in Eqs. (A.1)–

(A.3) belongs to a particular problem. It is implied thatsuch specification should meet certain conditions of al-gebraic consistency. For example, summation in (A.1)should be consistent with the “Kronecker symbol” in(A.3), (with fκ being an arbitrary function of the modeindex)

∑

κ′

δκκ′fκ′ = fκ. (A.4)

Similar consistencies should exist between integrationsand corresponding delta-functions,

∫

dτ ′δ(τ, τ ′)f(τ ′) = f(τ),

∫

dξ′δ(ξ, ξ′)f(ξ′) = f(ξ).

(A.5)

The delta-functions and Kronecker symbols are symmet-ric. These consistencies extend to functional differenti-ations by c-number functions of ξ and τ . So, for thec-number functions aκ(τ), aκ(τ) occuring in Eqs. (A.14),(A.15),

δaκ(τ)

δaκ′(τ ′)=

δaκ(τ)

δaκ′(τ ′)= δ(τ, τ ′)δκκ′ ,

δF (a)

δaκ(τ)=

δF (a)

δaκ(τ)= 0,

(A.6)

and similarly for the c-number pair X(ξ, τ), X(ξ, τ) in(A.10), (A.11). Conditions (A.4)–(A.6) make all rela-tions below algebraically defined.Linear order of the generalised time axis allows one to

define the step-function,

θ(τ, τ ′) =

{

1, τ ≻ τ ′,

0, τ ≺ τ ′,(A.7)

and the “time” ordering,

T X(ξ, τ) ˆX(ξ, τ) = θ(τ, τ ′)X(ξ, τ) ˆX(ξ, τ)

+ θ(τ ′, τ) ˆX(ξ, τ)X(ξ, τ), (A.8)

cf. Eq. (46). Similar definitions apply to a larger num-ber of factors. The time-symmetric ordering is definedreplacing “larger than” by “succeed” in Eq. (20):

T WX1(τ1)X2(τ2) · · · Xn(τn) =1

2N−1

×[

· · ·[[

X1(τ1), X2(τ2)]

+, X3(τ3)

]

+, · · · , XN (τN )

]

+,

τ1 ≻ τ2 ≻ · · · ≻ τN . (A.9)

The proof of section IID that for the free operators sym-metric and time-symmetric orderings are the same thinggeneralises literally with t → τ and ω0t → ϕκ(τ).

The generalised structural symmetric Wick theoremthat we now wish to prove reads,

TP [X, ˆX ] = T W

{

expZ[

δ

δX,

δ

δX

]

×P [X, X]|X→X,X→ ˆX

}

,

(A.10)

where X(ξ, τ), X(ξ, τ) are a pair of c-number functionsand,

Z[

δ

δX,

δ

δX

]

= −ih

∫

dτdτ ′dξdξ′

×GW (ξ, τ ; ξ′, τ ′)δ2

δX(ξ, τ)δX(ξ′, τ ′). (A.11)

Applying (A.10) to T X(ξ, τ) ˆX(ξ′, τ ′) we find thatGW (ξ, τ ; ξ′, τ ′) coincides with the symmetric contractionof the pair (A.1),

− ihGW (ξ, τ ; ξ′, τ ′)

= T X(ξ, τ) ˆX(ξ′, τ ′)− T WX(ξ, τ) ˆX(ξ′, τ ′). (A.12)

Since all operator reorderings occur mode-wise, it suf-fices to verify (A.10) for one mode; the general formulathen follows by direct calculation. In the one-mode case,

X(ξ, τ) → aκ(τ),ˆX(ξ, τ) → a†κ(τ),

GW (ξ, τ ; ξ′, τ ′) → GWκ (τ, τ ′)

= i[

T aκ(τ)a†κ(τ

′)− T Waκ(τ)a†κ(τ

′)]

=i

2e−iϕκ(τ)+iϕκ(τ

′)[

θ(τ, τ ′)− θ(τ ′, τ)]

, (A.13)

and Eq. (A.10) becomes,

TP [a, a†] = T W

{

expZκ

[

δ

δa,δ

δa

]

× P [a, a]|a→aκ,a→a†κ

}

, (A.14)

where a(τ), a(τ) are a pair of arbitrary c-number func-tions of “time.” The “per mode” reordering exponent Zκ

reads,

Zκ

[

δ

δa,δ

δa

]

= −i

∫

dτdτ ′GWκ (τ, τ ′)

δ2

δa(τ)δa(τ ′).

(A.15)

Verification of Eq. (A.14) goes in two steps. Firstly, wenote that the equivalence between the verbal and alge-braic forms of Wick’s theorem proper [52] does not de-pend on details of the contraction and hence equally ap-plies to the symmetric Wick theorem. Secondly, the proofof the symmetric Wick theorem for the time axis in sec-tion III B may be literally adjusted to generalised time

20

replacing time t by τ , the symbol > by ≻, and the con-traction GW

++(t − t′) by GW (τ, τ ′). We therefore regardEq. (A.14) proven.To verify the general relation (A.10) we note that

any functional depending on the mode operators may bebrought to the symmetric form by applying Eq. (A.14)mode-wise:

TP [X, ˆX ] = T W

(

exp

{

∑

κ

(

Zκ

[

δ

δaκ,

δ

δaκ

]

+ Zκ

[

δ

δbκ,

δ

δbκ

])}

P [X, X]|a→a

)

. (A.16)

In this formula, the c-number fields X(ξ, τ), X(ξ, τ) aregiven by Eq. (A.1) without hats,

X(ξ, τ) =√h∑

κ

[

uκ(ξ)aκ(τ) + vκ(ξ)bκ(τ)]

,

X(ξ, τ) =√h∑

κ

[

uκ(ξ)aκ(τ) + vκ(ξ)bκ(τ)]

,(A.17)

where aκ(τ), aκ(τ), bκ(τ), and bκ(τ) are four independentc-number fields per mode, and a → a is a shorthandnotation for the substitution,

aκ(τ) → aκ(τ), aκ(τ) → a†κ(τ),

bκ(τ) → bκ(τ), bκ(τ) → b†κ(τ).(A.18)

The “per mode” reordering exponents are given by(A.15). Applying the chain rule to differentiations in(A.16) we get,

δ

δa(τ)P [X, X] =

√h

∫

dξ uκ(ξ)δ

δX(ξ, τ)P [X, X],

δ

δa(τ)P [X, X] =

√h

∫

dξ uκ(ξ)δ

δX(ξ, τ)P [X, X],

δ

δb(τ)P [X, X] =

√h

∫

dξ vκ(ξ)δ

δX(ξ, τ)P [X, X],

δ

δb(τ)P [X, X] =

√h

∫

dξ vκ(ξ)δ

δX(ξ, τ)P [X, X].

(A.19)

Substituting these relations into Eq. (A.16) we recover

Eq. (A.10) with

GW (ξ, τ ; ξ′, τ ′) =∑

κ

[

uκ(ξ)uκ(ξ′)GW

κ (τ, τ ′)

+ vκ(ξ′)vκ(ξ)G

Wκ (τ ′, τ)

]

. (A.20)

It is readily verified that this definition of GW (ξ, τ ; ξ′, τ ′)agrees with (A.12). The only remaining artifact ofthe underlying mode expansion is then the substitution(A.18). However, with everything expressed in terms ofX(ξ, τ), X(ξ, τ), it may be replaced by

X(ξ, τ) → X(ξ, τ), X(ξ, τ) → ˆX(ξ, τ). (A.21)

This concludes the proof of Eq. (A.10).

We conclude this appendix with a remark on scalingof fields and contractions in the classical limit h → 0.Commutators of physical quantities must scale as h. Inthis sense, quantised mode amplitudes aκ are not physi-cal. The physical amplitude is

√haκ, which in the limit

h → 0 becomes the classical mode amplitude [53]. For

this reason the factor√h is explicitly present in the defi-

nitions of “physical fields” (A.1). The scaling ∝ h is thenremoved from the contraction (A.12) by the factor h onthe LHS. The result is that, as was demonstrated in Refs.[32, 33], Green functions (propagators) of bosonic fieldsare to a large extent classical quantities. In particular,they are all expressed by the response transformation interms of the retarded Green function of the correspond-ing classical field. For more details see [32–34, 53].

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